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According to the Knaster conjecture, for any continuous function f:Sm+n−2→Rm and n distinct points u1,u2,…,un∈Sm+n−2, there exists a rotation r∈SO(m+n−1) such that f(ru1)=f(ru2)=⋯=f(run). In this paper, we focus on the study of the properties of a continuous mapping from a sphere to a Euclidean space by using the theory of Smith periodic transformation and the Smith special index of the Stiefel manifold...
Given a continuous function f:Sm+n−2→Rm, and n points u1,u2,…,un∈Sm+n−2; does there exist a rotation r∈SO(m+n−1) such that f(ru1)=f(ru2)=⋯=f(run)? In this paper, we study the property of a continuous map from a sphere to a Euclidean space by using the theory of Smith periodic transformation and Brouwer degree of map theorem. The conjecture is proved under the case of n=2 and m being even. Furthermore,...
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